Example H6.3

MATLAB code for example 6.3 from the book "Regeltechniek voor het HBO"

Asymptotes of a root locus

Date : 02/03/2022
Revision : 1.0

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Asymptotes of a root locus Asymptotes of a root locus of 3 poles and one zero MATLAB code for this example

Asymptotes of a root locus of 3 poles and one zero

Suppose the transfer function

in figure 6.4 is:

From example 6.2 we already know that

and

are the starting points of the three root loci branches in the s-plane. One branche will run to the end point in

, and the remaining two branches thus run - each along an asymptote - to a point infinitely far away from the origin in the s-plane.

So there are two asymptotes that intersect in the point q on the real axis in the so called center of gravity of the pole and zero plot. According to formula 6.13 we get:

In this expression the meaning of n is the number of starting points (often determined by the number of poles of the open system) and m the number of finite ending points (often determined by the number of zeros of the open system). But watch out: when poles and/or zeros are present in the feed back loop, you have to add them to the poles and zeros of the open sytem! So if you start from the start and end points, you never make a mistake.

The angles

at which the braches intersect the horizontal real axis are (formula 6.13):

with

. In this case

and 1.

So the angles under which the asymptotes intersect the real axis are:

rad

and

rad

. (remember that counterclockwise rotation is the positive direction in the complex plane). We should note that these asymptotes divide 360 degrees into equal parts.

Note that formula 6.13 and 6.16 apply to positive values of

, which is almost always the case.

is negative, then in formula 6.7 holds:

and formula 6.13 becomes:

In this book we always assume

is positive, unless it is explicitly stated that it is not.

MATLAB code for this example

% clear all variables from Workspace and close all figures.

clear variables;

close all;

% Define 's' variable

s=tf('s');

% Determine parameters.

K=1;

p1 = 3;

p2 = 5;

z1 = 2;

% Create the 2nd order proces

H=K*(s+z1)/(s*(s+p1)*(s+p2));

figure(101);

% Show the root locus plot of H

rlocus(H);